A note on the relation between strong and M-stationarity for a class of mathematical programs with equilibrium constraints

René Henrion; Jiří Outrata; Thomas Surowiec

Kybernetika (2010)

  • Volume: 46, Issue: 3, page 423-434
  • ISSN: 0023-5954

Abstract

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In this paper, we deal with strong stationarity conditions for mathematical programs with equilibrium constraints (MPEC). The main task in deriving these conditions consists in calculating the Fréchet normal cone to the graph of the solution mapping associated with the underlying generalized equation of the MPEC. We derive an inner approximation to this cone, which is exact under an additional assumption. Even if the latter fails to hold, the inner approximation can be used to check strong stationarity via the weaker (but easier to calculate) concept of M-stationarity.

How to cite

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Henrion, René, Outrata, Jiří, and Surowiec, Thomas. "A note on the relation between strong and M-stationarity for a class of mathematical programs with equilibrium constraints." Kybernetika 46.3 (2010): 423-434. <http://eudml.org/doc/196667>.

@article{Henrion2010,
abstract = {In this paper, we deal with strong stationarity conditions for mathematical programs with equilibrium constraints (MPEC). The main task in deriving these conditions consists in calculating the Fréchet normal cone to the graph of the solution mapping associated with the underlying generalized equation of the MPEC. We derive an inner approximation to this cone, which is exact under an additional assumption. Even if the latter fails to hold, the inner approximation can be used to check strong stationarity via the weaker (but easier to calculate) concept of M-stationarity.},
author = {Henrion, René, Outrata, Jiří, Surowiec, Thomas},
journal = {Kybernetika},
keywords = {mathematical programs with equilibrium constraints; S-stationary points; M-stationary points; Fréchet normal cone; limiting normal cone; mathematical programs with equilibrium constraints; S-stationary points; M-stationary points; Fréchet normal cone; limiting normal cone},
language = {eng},
number = {3},
pages = {423-434},
publisher = {Institute of Information Theory and Automation AS CR},
title = {A note on the relation between strong and M-stationarity for a class of mathematical programs with equilibrium constraints},
url = {http://eudml.org/doc/196667},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Henrion, René
AU - Outrata, Jiří
AU - Surowiec, Thomas
TI - A note on the relation between strong and M-stationarity for a class of mathematical programs with equilibrium constraints
JO - Kybernetika
PY - 2010
PB - Institute of Information Theory and Automation AS CR
VL - 46
IS - 3
SP - 423
EP - 434
AB - In this paper, we deal with strong stationarity conditions for mathematical programs with equilibrium constraints (MPEC). The main task in deriving these conditions consists in calculating the Fréchet normal cone to the graph of the solution mapping associated with the underlying generalized equation of the MPEC. We derive an inner approximation to this cone, which is exact under an additional assumption. Even if the latter fails to hold, the inner approximation can be used to check strong stationarity via the weaker (but easier to calculate) concept of M-stationarity.
LA - eng
KW - mathematical programs with equilibrium constraints; S-stationary points; M-stationary points; Fréchet normal cone; limiting normal cone; mathematical programs with equilibrium constraints; S-stationary points; M-stationary points; Fréchet normal cone; limiting normal cone
UR - http://eudml.org/doc/196667
ER -

References

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  1. Dontchev, A. L., Rockafellar, R. T., 10.1137/S1052623495284029, SIAM J. Optim. 6 (1996), 1087–1105. Zbl0899.49004MR1416530DOI10.1137/S1052623495284029
  2. Dontchev, A. L., Rockafellar, R. T., 10.1137/S1052623400371016, SIAM J. Optim. 12 (2001), 170–187. Zbl1008.49009MR1870590DOI10.1137/S1052623400371016
  3. Henrion, R., Jourani, A., Outrata, J. V., 10.1137/S1052623401395553, SIAM J. Optim. 13 (2002), 603–618. Zbl1028.49018MR1951037DOI10.1137/S1052623401395553
  4. Henrion, R., Römisch, W., 10.1007/s10492-007-0028-z, Appl. Math. 52 (2007), 473–494. MR2357576DOI10.1007/s10492-007-0028-z
  5. Henrion, R., Outrata, J. V., Surowiec, T., 10.1016/j.na.2008.11.089, Nonlinear Anal. 71 (2009), 1213–1226. MR2527541DOI10.1016/j.na.2008.11.089
  6. Henrion, R., Outrata, J. V., Surowiec, T., Analysis of M-stationary points to an EPEC modeling oligopolistic competition in an electricity spot market, Weierstraß-Institute of Applied Analysis and Stochastics, Preprint No. 1433 (2009) and submitted. MR2527541
  7. Henrion, R., Mordukhovich, B. S., Nam, N. M., Second-order analysis of polyhedral systems in finite and infinite dimensions with applications to robust stability of variational inequalities, SIAM J. Optim., to appear. MR2650845
  8. Luo, Z.-Q., Pang, J.-S., Ralph, D., Mathematical Programs with Equilibrium Constraints, Cambridge University Press, Cambridge 1996. Zbl1139.90003MR1419501
  9. Mordukhovich, B. S., Approximation Methods in Problems of Optimization and Control (in Russian), Nauka, Moscow 1988. MR0945143
  10. Mordukhovich, B. S., Variational Analysis and Generalized Differentiation, Vol. 1: Basic Theory. Springer, Berlin 2006. 
  11. Mordukhovich, B. S., Variational Analysis and Generalized Differentiation, Vol. 2: Applications. Springer, Berlin 2006. MR2191745
  12. Flegel, M. L., Kanzow, C., Outrata, J. V., 10.1007/s11228-006-0033-5, Set-Valued Anal. 15 (2007), 139–162. Zbl1149.90143MR2321948DOI10.1007/s11228-006-0033-5
  13. Outrata, J. V., Kočvara, M., Zowe, J., Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, Kluwer Academic Publishers, Dordrecht 1998. MR1641213
  14. Pang, J.-S., Fukushima, M., 10.1023/A:1008656806889, Comput. Optim. Appl. 13 (1999), 111–136. MR1704116DOI10.1023/A:1008656806889
  15. Robinson, S. M., 10.1007/BFb0120929, Math. Program. Studies 14 (1976), 206–214. Zbl0449.90090MR0600130DOI10.1007/BFb0120929
  16. Robinson, S. M., 10.1287/moor.5.1.43, Math. Oper. Res. 5 (1980), 43–62. Zbl0437.90094MR0561153DOI10.1287/moor.5.1.43
  17. Rockafellar, R. T., Wets, R. J.-B., Variational Analysis, Springer, Berlin 1998. Zbl0888.49001MR1491362
  18. Scheel, H., Scholtes, S., 10.1287/moor.25.1.1.15213, Math. Oper. Res. 25 (2000), 1–22. MR1854317DOI10.1287/moor.25.1.1.15213
  19. Surowiec, T., Explicit Stationarity Conditions and Solution Characterization for Equilibrium Problems with Equilibrium Constraints, Doctoral Thesis, Humboldt University, Berlin 2009. 

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